Electrometers measure a wide range of currents at extremely low levels. For example, mass spectrometers use current sensors having eight decades of coverage ranging from less than 0.1 picoamps (pA) to 10.0 femtoamps (fA). The most straightforward way to measure current is to use an operational amplifier in the feedback configuration. FIG. 1 illustrates a feedback configuration in which resistor R (101) is the feedback element. The operational amplifier 30 includes an inverting input 32A, a non-inverting input 32B, and an output 34. Resistor R is electrically coupled between the output 34 of the amplifier 30 and the inverting input 32A of the amplifier 30 to form a feedback loop. The non-inverting input 32B is grounded. In this configuration, the output voltage Vout is related to the input current Iin according to the following relationship:Vout=IinR.
Such a configuration, however, suffers from a number of problems. For a given bandwidth and temperature, the input current noise is inversely proportional to the square root of the resistance of resistor R. Therefore, as the resistance of resistor R decreases, the input current noise increases (similarly, as the resistance of resistor R increases, the input current noise decreases).
The input current noise is also proportional to the square root of the bandwidth and the absolute temperature. The bandwidth, which determines the response time, is dominated by the time constant formed from the value of the feedback resistor R and its inherent capacitance as well as any stray capacitance. To improve the response time, the time constant can be reduced by choosing a resistor R with a small resistance value. But this has the effect of increasing the noise partly because of the increased bandwidth and partly because of a smaller resistance.
Integration can be used to reduce the noise. The noise is inversely proportional to the square root of the integration time. Therefore, as the integration time increases, the noise decreases. But integration effectively reduces the bandwidth and increases the response time of the electrometer. The relationship between integration time and bandwidth is
  Bw  =            1              π        ·        T              ⁢                  ⁢          Hz      .      Thus, a one-second integration time is equivalent to a bandwidth of 0.318 Hz. And as the integration time increases, bandwidth decreases thereby increasing the response time.
An electrometer that utilizes an operational amplifier with a single feedback resistor R also suffers from a limited dynamic range. The dynamic range is determined by the ratio of the largest signal which can be measured to twice the RMS noise level, which may represent the smallest signal that can be adequately detected. For a full-scale signal of 10 Volts, the full-scale current will be
      10    R    ⁢          ⁢      Amps    .  The voltage noise across the resistor R is given by √{square root over (4·K·T·R·Bw)} Volts RMS, where K=1.38×10−23 (Boltzman's constant), T=absolute temperature in ° K, R=resistance in Ohms, and Bw=Bandwidth in Hertz (assuming a brick-wall filter). Therefore the equivalent current noise will be
                    4        ·        K        ·        T        ·        R        ·        Bw              R    ⁢  Amps  ⁢          ⁢      RMS    .  
If the minimum detectable signal is twice the noise then the dynamic range will be
      5                  4        ·        K        ·        T        ·        R        ·        Bw              :  1.Table 1 below lists the dynamic ranges obtainable from various resistors and bandwidths at an absolute temperature of 300° K:
TABLE 1Resistorvalues inBandwidthohms1 Hz10 Hz100 Hz1 KHz 50 M5.49 × 1061.74 × 1065.49 × 1051.74 × 105100 M3.89 × 1061.25 × 1063.89 × 1051.23 × 105 1 G1.23 × 1063.89 × 1051.23 × 1053.89 × 104 10 G3.89 × 1051.23 × 1053.89 × 1041.23 × 104100 G1.23 × 1053.89 × 1041.23 × 1043.89 × 103
As shown in Table 1, the best dynamic range is achieved with resistors having lower resistances. In some applications the maximum available signal current is around 200 nA, which for a 10 Volt signal represents a resistor of 50 megaohms. However, a feedback resistor R with a resistance value lower than 50 megaohms cannot be used to obtain a better dynamic range because the full-scale output would not be possible. Indeed, if a large input current were applied to a feedback resistor R with a resistance value lower than 50 megaohms, the voltage output Vout would saturate (i.e., Vout will be equal to the source voltage levels applied to the operational amplifier 30).
One way to increase the dynamic range is to add a resistor bank including resistors Ra, Rb, and Rc (102a-c) and switches or relays Sa, Sb, and Sc (103a-c) (as shown in FIG. 1) that provide selectable resistance levels. Such a resistor bank, however, has several disadvantages. First, switches with minimal leakage current are sophisticated and therefore expensive. Second, switches with relays induce electrostatic fields and introduce other disturbances that interfere with the input current signal. Third, the use of switches results in lengthy settling times when switching between resistors. Finally, switches suffer from overload recovery time problems if a large over-scale input signal is applied to a resistor with a high-resistance value.